Which equation could be solved using this application of the quadratic formula?
[tex]x=\frac{-8 \pm \sqrt{8^2-4(3)(-2)}}{2(3)}[/tex]
Answer (2)
Compare the given quadratic formula with the general form.
Identify the coefficients: a = 3 , b = 8 , c = − 2 .
Substitute the coefficients into the general quadratic equation a x 2 + b x + c = 0 .
The quadratic equation is 3 x 2 + 8 x − 2 = 0 .
Explanation
Problem Analysis The problem provides a quadratic formula application and asks us to identify the corresponding quadratic equation. The given quadratic formula is:
General Quadratic Formula x = 2 ( 3 ) − 8 \tpm 8 2 − 4 ( 3 ) ( − 2 )
We need to compare this with the general quadratic formula:
Formula Comparison x = 2 a − b \tpm b 2 − 4 a c
Identifying Coefficients By comparing the given formula with the general formula, we can identify the coefficients:
Coefficients Values a = 3 , b = 8 , and c = − 2 .
Substituting into General Equation Now, we substitute these values into the general quadratic equation:
General Quadratic Equation a x 2 + b x + c = 0
Final Equation Substituting a = 3 , b = 8 , and c = − 2 , we get:
Resulting Quadratic Equation 3 x 2 + 8 x − 2 = 0
Examples
Quadratic equations are used in various real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, and modeling the growth or decay of populations. For example, if you want to build a rectangular garden with an area of 100 square meters and a perimeter of 40 meters, you can use a quadratic equation to find the length and width of the garden. The equation helps to optimize the use of space and resources.
The equation that can be solved using the given quadratic formula is 3 x 2 + 8 x − 2 = 0 . By identifying the coefficients from the formula, we constructed this quadratic equation. This equation illustrates a variety of applications in fields such as physics and optimization.
;
